Obesity rates are soaring in Saudi Arabia and other wealthy Gulf states, leading to a boom in bariatric surgery on children.

RIYADH, Saudi Arabia—Daifailluh al-Bugami was just a year old when his parents noticed that his lips turned blue as he slept at night. It was his weight, doctors said, putting pressure on his delicate airways.

Now Daifailluh is 3, and at 61 pounds he is nearly double the typical weight of a child his age. So the Bugamis are planning the once unthinkable: To have their toddler undergo bariatric surgery to permanently remove part of...